Valuation of Stock Option Using Balck-Scholes Model - Part 3

May 19, 2026

3.1 Introduction of the BlackScholes–Merton Model

In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton achieved a major breakthrough in financial economics by developing what is now known as the Black–Scholes–Merton (BSM) model. This framework completely revolutionized how traders price and hedge derivatives. In 1997, the importance of this model was recognized with the Nobel Prize in Economics awarded to Merton and Scholes (Fischer Black passed away in 1995, otherwise he too would have been a recipient).

While previous researchers had correctly calculated the expected future payoff of a European option, they struggled to find the correct discount rate to use, since an option's risk profile changes continuously over time. The creators of the model solved this problem using two distinct insights:

The Black–Scholes Approach: They used the Capital Asset Pricing Model (CAPM) to determine a dynamic relationship between the required return on the option and the required return on the stock.
The Merton Approach: Merton set up a riskless portfolio consisting of the option and the underlying stock. He argued that because the portfolio is completely risk-free over a short period of time, it must earn exactly the risk-free rate of return to prevent arbitrage.

Merton's approach is more general because it does not rely on CAPM assumptions. Part 3 covers his dynamic hedging approach to deriving the model, explains how to estimate historical and implied volatility, applies risk-neutral valuation, and extends the formula to dividend-paying stocks.

3.2 Lognormal Disctribution of Stock Prices

The foundational assumption of the Black–Scholes–Merton model is that percentage changes in a stock price over a very short period of time ($\Delta t$) follow a normal distribution. To analyze this behavior, we define the following annualized parameters:

$m$: The expected rate of return on the stock per year.
$s$: The volatility of the stock price per year.

Recall from our previous work in Part 2 that the discrete-time return framework over a short interval $\Delta t$ can be expressed using equation (2.1):

$$\frac{\Delta S}{S} \sim \phi(\mu \, \Delta t, \sigma^2 \Delta t) $$

where $\Delta S$ is the change in the stock price $S$ in time $\Delta t$, and $\phi(m, v)$ denotes a normal distribution with mean $m$ and variance $v$.

Similarly, we can recall the continuous long-term behavior of the log stock price over a full time horizon $T$ from equation (2.3) in Part 2. The equation implies that

$$\ln S_T - \ln S_0 \sim \phi\left( \left( \mu - \frac{1}{2} \sigma^2 \right) T, \,\, \sigma^2 T \right) $$

Using the basic properties of logarithms, we can rewrite this relationship in alternative forms to isolate either the log return ratio or the final log price distribution.

First, because $\ln S_T - \ln S_0 = \ln \left( \frac{S_T}{S_0} \right)$, the continuously compounded return ratio over a horizon $T$ is distributed as:

$$\ln \frac{S_T}{S_0} \sim \phi \left[ \left( \mu - \frac{\sigma^2}{2} \right) T, \sigma^2 T \right] $$

Alternatively, by adding $\ln S_0$ to both sides of the distribution parameters, we can isolate the terminal log stock price at time $T$:

$$\ln S_T \sim \phi \left[ \ln S_0 + \left( \mu - \frac{\sigma^2}{2} \right) T, \sigma^2 T \right] \tag{3.1} $$

This final expression confirms that the natural logarithm of the terminal stock price is normally distributed. Consequently, the absolute stock price $S_T$ itself follows a lognormal distribution.

3.2.1 The Expected Value and Variance of $S_T$

To find the expected value and variance of the absolute terminal stock price $S_T$, we utilize the mathematical properties of a lognormal distribution. If a random variable $X$ is normally distributed such that $X \sim \phi(\mu, \sigma^2)$, then its exponentiated form $Y = e^X$ is lognormally distributed and follows these standard statistical definitions:

Expected Value: $E(Y) = e^{\mu + \frac{1}{2}\sigma^2}$
Variance: $\text{var}(Y) = e^{2\mu + \sigma^2} \left( e^{\sigma^2} - 1 \right)$

From equation (3.1), our underlying log stock price random variable has the following parameters:

$E(\ln S_T) = \ln S_0 + \left( m - \frac{\sigma^2}{2} \right) T$
$\text{var}(\ln S_T) = \sigma^2 T$

Plugging our specific log price parameters ($\mu$ and $\sigma^2$) into the lognormal expected value shortcut yields:

$$E(S_T) = e^{\left[ \ln S_0 + \left( m - \frac{\sigma^2}{2} \right) T \right] + \frac{1}{2}(\sigma^2 T)}$$

We can expand the terms inside the exponent algebraically:

$$E(S_T) = e^{\ln S_0} \cdot e^{\mu T} \cdot e^{-\frac{1}{2}\sigma^2 T} \cdot e^{\frac{1}{2}\sigma^2 T}$$

Because $e^{\ln S_0} = S_0$ and the volatility exponents cancel each other out completely ($e^{-\frac{1}{2}\sigma^2 T + \frac{1}{2}\sigma^2 T} = e^0 = 1$), the equation simplifies directly to:

$$E(S_T) = S_0 e^{\mu T} \tag{3.2} $$

Next, we plug our specific log price parameters ($\mu$ and $\sigma^2$) into the lognormal variance shortcut:

$$\text{var}(S_T) = e^{2\left[ \ln S_0 + \left( \mu - \frac{\sigma^2}{2} \right) T \right] + \sigma^2 T} \left( e^{\sigma^2 T} - 1 \right)$$

Distributing the factor of $2$ across the first bracket gives:

$$\text{var}(S_T) = e^{2\ln S_0 + 2\mu T - \sigma^2 T + \sigma^2 T} \left( e^{\sigma^2 T} - 1 \right)$$

The $-\sigma^2 T$ and $+\sigma^2 T$ terms inside the exponent cancel out perfectly. Separating the remaining exponential elements results in:

$$\text{var}(S_T) = e^{\ln(S_0^2)} \cdot e^{2\mu T} \left( e^{\sigma^2 T} - 1 \right)$$

Since $e^{\ln(S_0^2)} = S_0^2$, we arrive at the final expression for the variance of the terminal stock price:

$$\text{var}(S_T) = S_0^2 e^{2m T} \left( e^{\sigma^2 T} - 1 \right) \tag{3.3}$$

3.3 Derivative ($f$) using Itô's Lemma

Suppose that $f$ is the price of a call option or another derivative instrument whose value is contingent on the underlying stock price $S$. Because the contract value changes as both the stock price moves and time passes, the variable $f$ must be a continuous function of both $S$ and $t$, written as $f(S, t)$.

To track how the value of this derivative changes over an infinitesimal time step, we recall our general presentation of Itô's Lemma from equation (2.2) in Part 2:

$$dG = \left( a \frac{\partial G}{\partial S} + \frac{\partial G}{\partial t} + \frac{1}{2} b^2 \frac{\partial^2 G}{\partial S^2} \right) dt + b \frac{\partial G}{\partial S} \, dz \tag{2.2}$$

In our framework, the underlying asset $S$ follows geometric Brownian motion where the absolute drift parameter is $a = \mu S$ and the absolute volatility parameter is $b = \sigma S$. Replacing the general function $G$ with our option price function $f$, we find the continuous-time process for the derivative:

$$df = \left( \frac{\partial f}{\partial S} \mu S + \frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \right) dt + \frac{\partial f}{\partial S} \sigma S \, dz \tag{3.4}$$

The discrete change in the value of our contingent derivative ($\Delta f$) over that same short interval $\Delta t$ is approximated as:

$$\Delta f = \left( \frac{\partial f}{\partial S} \mu S + \frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \right) \Delta t + \frac{\partial f}{\partial S} \sigma S \Delta z \tag{3.5} $$

where $\Delta f$ and $\Delta S$ are the absolute changes in the option price and stock price respectively, and $\Delta z$ represents a discrete Wiener random shock ($\epsilon\sqrt{\Delta t}$).

3.4 Construction of the Delta-Hedging Portfolio

From our discrete-time equations for the stock ($\Delta S$) and the derivative ($\Delta f$), we notice a vital reality: the underlying random Wiener shock ($\Delta z$) is identical in both equations. Because the derivative's randomness is entirely driven by the underlying stock's randomness, we can construct a combined portfolio of the stock and the derivative in a specific ratio that completely eliminates the Wiener process shock.

We define $\Pi$ as the total value of this specific portfolio. The construction rules are:

$-1$: Short one derivative contract
$+\frac{\partial f}{\partial S}$: Long shares of the underlying stock

Therefore, by definition, the total accounting value of this portfolio is written as:

$$\Pi = -f + \frac{\partial f}{\partial S}S \tag{3.6}$$

Let's first clarify two essential financial trading terms used here:

Short: Selling an asset you do not own, or taking on a liability/obligation that you must repay later. This is represented as a negative sign ($-$) in financial equations.
Long: Buying and holding an asset in your possession. This is represented as a positive sign ($+$) in financial equations.

In plain English, this strategy means: "We write and sell exactly one derivative contract ($-f$), and to protect ourselves from its price movements, we simultaneously purchase a specific fraction of the underlying stock ($+\frac{\partial f}{\partial S}S$)."

Think of $\Pi$ (Pi) as the net balance sheet value of this combined portfolio:

$-f$: The value of the single derivative contract we shorted. If the option price goes up, our liability grows, making this a negative position on our balance sheet. (The letter $f$ stands for financial derivative instrument).
$S$: The current spot price of the underlying stock.
$\frac{\partial f}{\partial S}$: The exact number of shares we hold. This partial derivative measures how many dollars the option price ($f$) moves for every \$1 change in the stock price ($S$). In quantitative finance, this vital sensitivity metric is known as 'Delta ($\Delta$)'.

Why do we buy exactly $\frac{\partial f}{\partial S}$ shares?

Imagine we sold a call option that has a Delta ($\frac{\partial f}{\partial S}$) of exactly $0.5$. This tells us: "If the stock price rises by \$1.00, the option price will rise by \$0.50."

If we are short this option ($-f$), a sudden \$1.00 spike in the stock price increases our option liability by \$0.50, causing us a \$0.50 loss. To perfectly neutralize (hedge) this risk, we need a long stock position that generates an identical \$0.50 gain under the exact same conditions. Since a whole share gains \$1.00, we simply buy exactly 0.5 shares ($\frac{\partial f}{\partial S}$ shares).

Net Portfolio Outcome if the Stock Price Rises by \$1.00:
- Loss from the short option position: $-\$0.50$
- Gain from the long stock position: $+\$0.50$ ($\$1.00 \text{ price increase} \times 0.5 \text{ shares}$)
- Net Portfolio Change: \$0.00 (Risk is completely eliminated!)

Over a very short time interval $\Delta t$, the change in the absolute value of our portfolio ($\Delta \Pi$) is determined by the discrete changes in its components:

$$\Delta \Pi = -\Delta f + \frac{\partial f}{\partial S}\Delta S \tag{3.7}$$

Recall the discrete version of equation (2.1) in Part 2:

$$\Delta S = \mu S \, \Delta t + \sigma S \, \Delta z \tag{3.8}$$

Now, we substitute our discrete-time asset formulas—equation (3.8) for $\Delta S$ and equaton (3.5) for $\Delta f$—into equation (3.7). The framework expands to:

$$\Delta \Pi = -\left[ \left( \frac{\partial f}{\partial S} \mu S + \frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \right) \Delta t + \frac{\partial f}{\partial S} \sigma S \Delta z \right] + \frac{\partial f}{\partial S} \left[ \mu S \Delta t + \sigma S \Delta z \right]$$

When you distribute the negative sign and expand the equation, a remarkable cancellation happens. The terms containing the drift ($\frac{\partial f}{\partial S} \mu S \Delta t$) and, most importantly, the terms containing the random Wiener shock ($\frac{\partial f}{\partial S} \sigma S \Delta z$) cancel out completely.

This leaves us with a clean, deterministic equation for the change in portfolio value:

$$\Delta \Pi = -\left(\frac{\partial f}{\partial t} + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \right) \Delta t \underbrace{- \frac{\partial f}{\partial S} \mu S \Delta t + \frac{\partial f}{\partial S} \mu S \Delta t}_{\text{Cancels Out!}} \underbrace{- \frac{\partial f}{\partial S} \sigma S \Delta z + \frac{\partial f}{\partial S} \sigma S \Delta z}_{\text{The Magic Part: Risk Vanishes!}}$$

$$\Delta \Pi = \left( -\frac{\partial f}{\partial t} - \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \right) \Delta t \tag{3.9}$$

Because equation (3.9) does not involve the random variable $\Delta z$, the portfolio must be completely riskless during the short time interval $\Delta t$. In an efficient market with no arbitrage opportunities, a completely riskless portfolio must instantaneously earn the exact same rate of return as other short-term risk-free securities ($r$). If it earned more than $r$, arbitrageurs would secure a riskless profit by borrowing money at the risk-free rate to buy this portfolio. If it earned less, they would short the portfolio to buy risk-free bonds.

Therefore, the return on the portfolio must follow this exact relationship:

$$\Delta \Pi = r \Pi \Delta t \tag{3.10}$$

By substituting our portfolio definition equation (3.10) and our change equation (3.12) into this arbitrage constraint, we obtain:

$$\left( -\frac{\partial f}{\partial t} - \frac{1}{2} \frac{\partial^2 f}{\partial S^2} \sigma^2 S^2 \right) \Delta t = r \left( -f + \frac{\partial f}{\partial S} S \right) \Delta t$$

We divide both sides by $\Delta t$ and move all derivative terms to the left-hand side:

$$\frac{\partial f}{\partial t} + r S \frac{\partial f}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 f}{\partial S^2} = r f \tag{3.11}$$

Equation (3.11) is the legendary Black–Scholes–Merton partial differential equation. It must be satisfied by any derivative security contingent on a non-dividend-paying stock $S$.